Methods for Spreading or Preventing Spreading of Information In a Network

ABSTRACT

A method for improved spreading of information in a network is described, together with corresponding methods with the opposite aim, namely to hinder the spreading of harmful information in a network. The harmful information may be (for example) a data virus. The first method includes as its characterizing feature to connect at least one node of high Eigenvector Centrality Index in a first region with at least one node of high Eigenvector Centrality Index in a second region. These connections may be made using direct links, or with the help of a new node lying between the nodes to be connected. One method for preventing spreading of information or physical traffic in a network may include as its characterizing feature to inoculate at least one centre node by blocking any transmission of unwanted information on all links in/out of said centre node. Another method for preventing spreading of information or physical traffic may be to inoculate all nodes in a ring of nodes surrounding a centre node by blocking any transmission of unwanted information on all links in/out of said nodes. Still another method may be to inoculate at least one bridge link connecting two regions by blocking any transmission of unwanted information on said link.

FIELD OF THE INVENTION

The present invention refers to a set of methods for managing networks(both logical and physical networks), within a number of areas. Moreparticularly the present invention discloses methods for spreading orpreventing spreading of information in a network, where the networkconsists of any number of network nodes connected by links. Theinventive methods are based on the method of analyzing networksdisclosed in Norwegian patent application NO 2003 5852; the content ofthis application is hereby incorporated by reference.

BACKGROUND OF THE INVENTION

It is a well known fact of this century that electronic information canspread to many people in a very short time. This fact is good news forsome people (spammers, bloggers), but can be rather bad news for thosepersons responsible for security. The battle against viruses, spam, andother forms of harmful or undesirable, self-propagating information isnever-ending.

In the following invention disclosure the problem of the spreading ofinformation has been approached from the direction of network analysis.The present invention includes both methods for helping information tospread more efficiently, and methods for hindering the spreading ofunwanted information (e.g., viruses). Much of the background discussionin the present invention disclosure is relevant to either purpose(helping desired information, or hindering unwanted information). Inthis document we will often use language (‘epidemic’, ‘infection’, etc)which is normally appropriate for the description of the spreading ofunwanted information. Our convention is however that thisepidemic-oriented language refers implicitly to both desired andundesired information, unless otherwise specified; it is used only forconvenience.

There are many kinds of models for epidemic spreading. In perhaps thesimplest class of such models, one assigns to each node only one of twopossible states: ‘uninfected’ or ‘infected’. If you are uninfected(‘susceptible’), you are deemed liable to be infected by any infectedneighbours. Correspondingly, if you are infected, you remain so for theduration of the experiment—and you remain capable of infecting any orall of your neighbours. Of course, on some appropriate time scale, nodesbecome ‘immune’ to the infection: a human develops antibodies, a machinegets antivirus software, the gossip becomes boring, or the innovationbecomes outmoded. We focus on a shorter time scale here, so that we canignore the state of acquired immunity. The technical name for our modelof spreading is ‘SI’, since the nodes have only two states: Susceptibleor Infected.

Since spreading takes place over the links of a network, it is clearthat the topology of the network can have a profound influence on thespreading process. In particular, we believe that the best understandingof spreading will come from a perspective which is based on a view ofthe whole network, and on an understanding of that network's structure.In earlier work [1], we have presented an approach to the analysis ofnetwork structure which is applicable to any network with symmetric(undirected) links. We also suggested that the analysis should be usefulfor the understanding of spreading over such a network. Recently [2], wehave developed a detailed semi-quantitative theory for how spreadingtakes place on such networks. The theory is based entirely on ourstructural analysis. The present invention addresses the question ofactive design or management of networks for the purpose of controlling(helping or hindering) spreading. Our analysis offers clear suggestionsfor how to control spreading in both of these senses.

Our approach departs from previous work in that we focus on both thetime and spatial progression of the epidemic spreading. We take aspatial resolution which is not microscopic, but rather at the level of‘neighbourhoods’-connected sub-graphs with roughly the same spreadingpower. More traditional approaches (reviewed in [4]) start from the‘well-mixed’ approximation, that every node can infect every other withsome probability, at all times. This approach may be said to have nonetwork perspective; or, it may be said to postulate a graph withextremely good mixing-such as a random graph of high degree, or acomplete graph. The review of Newman [4] also discusses more recentwork, involving a network perspective. All such work is based onwhole-graph properties, such as the node degree distribution; also,these approaches have focused on obtaining whole-graph results, eitherover time [5,6], or focusing especially on the infected fraction at verylong times [7]. This latter question is of course only interesting formodels more complex than the SI model; and indeed most work is directedtowards the behaviour of the SIS model (where nodes lose their infectionafter some time, and so become Susceptible again), or the SIR model(where nodes, after losing their infection, go through a refractoryperiod). Finally, we note that work analysing only whole-graphproperties cannot give the kinds of specific design improvements thatare embodied in the present invention.

Brauer [8] has examined the SI model for the case that the nodes(organisms, especially humans) are born and die. Because of the additionof these dynamic features, the steady infection rate is not necessarily100%. This work uses the well-mixed approximation, which gives rise tocoupled ordinary differential equations. Hence it too cannot suggestlocal, specific design improvements of the type included in the presentinvention.

A work which is perhaps closest to the present work is that of Wang etal [9]. Their model is SIS, in that nodes can be “cured”; but it isbased on a fully microscopic view of the network. In fact, their timeevolution operator is the same as that we develop in Ref. [2], with twodifferences. One is their addition of the “curing” term. This term issimply a multiple of the unit matrix, and so does not change thedominant eigenvector-which remains that of the adjacency matrix A.Because their model is SIS, the long-time infection fraction is notobvious, and must be solved for. The second difference in the timeevolution operator of Wang et al is that they neglect the crossterms—i.e. those arising from multiple transmissions to an infectednode. This approximation is valid for low infection fraction—while (aswe discuss below) it may also be good even as the infection fractionbecomes large. Wang et al report simulations which offer some supportfor this statement.

We emphasize that our work, like that of Wang et al [9], uses the fulladjacency matrix A in modelling the time evolution of the infection.Thus we start from a microscopic foundation. However, we will quicklyappeal to a ‘mesoscopic’ picture, in which it is meaningful and usefulto speak of neighbourhoods and their properties. As far as we know, ourwork is unique in this regard. This neighbourhood picture is the basisfor the methods (for improving the design of networks) which constitutethe present invention.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a method forimproved spreading of information in a network, and a correspondingmethod with the opposite aim, namely to hinder the spreading of harmfulinformation in a network.

These objects are achieved in the methods disclosed in the appendedpatent claims. In its first aspect, the present invention provides amethod for aiding the spreading of information or physical traffic in anetwork, said network including a number of network nodes interconnectedby links, including:

mapping the topology of the network,computing a value for link strength between the nodes,computing an Eigenvector Centrality index for all nodes, based on saidlink strength values,identifying nodes which are local maxima of the Eigenvector Centralityindex as centre nodes,grouping the nodes into regions surrounding each identified centre node,and with the characterizing feature of connecting at least one node ofhigh Eigenvector Centrality Index in a first region with at least onenode of high Eigenvector Centrality Index in a second region.

The second aspect of the invention relates to a method for spreadinginformation or physical traffic in a network, which includes as itscharacterizing feature to add at least one new node, and connecting atleast one existing node of high Eigenvector Centrality Index in each ofa first and second region with this said node.

In its third aspect, the invention relates to a method for preventingspreading of information or physical traffic in a network that ischaracterized in inoculating at least one node of high EigenvectorCentrality Index by blocking any transmission of unwanted information onall links in/out of said node.

In its fourth aspect, the invention relates to a method for preventingspreading of information or physical traffic in a network, said methodbeing characterized in inoculating at least one link of high EigenvectorCentrality Index connecting two regions by blocking any transmission ofunwanted information on said link.

BRIEF DESCRIPTION OF FIGURES

In order to make the invention more readily understandable, theinvention will now be discussed in detail in reference to theaccompanying figures, in which:

FIG. 1 is a schematic diagram showing the topographic view of a network,as used in the present discussion of the invention, with two ‘mountains’(regions).

FIG. 2 is a schematic picture showing epidemic spreading in one region.The two regions of FIG. 1 are now viewed from the ‘side’, as if theywere really mountains.

FIG. 3 is a schematic picture of the progress of an infection.

FIG. 4 is a schematic picture of connecting two high-EVC nodes indifferent regions with a link.

FIG. 5 is a schematic picture of connecting centres with a link.

FIG. 6 is a schematic picture of the procedure of inoculating the centrenodes in regions.

FIG. 7 is a schematic picture of the procedure of inoculating a ring(centred at the region centre node) of nodes one hop away from theregion centres.

FIG. 8 is a schematic picture of the procedure of inoculating high-EVCbridge nodes.

FIG. 9 is a schematic picture of the procedure of inoculating a high-EVCbridge link.

FIG. 10 is a schematic picture of two different ways of connecting twosubsets of nodes. In A, all the nodes in the one set are connected toall the nodes in the other set. In B, an extra node is inserted, andthis new node has connections to all or some of the nodes in each subsetof nodes. The new node has a star topology.

FIG. 11 is a schematic picture of two different ways of connecting a setof centre nodes. In A, all the centre nodes are connected to all othercentre nodes. In B, an extra node is inserted, and this new node hasconnections to all or a subset of the centre nodes. The new node has astar topology.

DETAILED DESCRIPTION OF THE INVENTION 1. Topography from Topology

An essential aspect of our approach to analysing the structure of anetwork is to define a measure of centrality for each node in thenetwork. There are in fact many different measures of centrality, mostof them coming from social science [10]. Our aim has been to find ameasure of centrality which implies well-connectedness. Furthermore, wewant a notion of well-connectedness which is not purely local. That is,we want a definition of well-connectedness (centrality) for node i whichtells us something about the neighbourhood of node i. We reason thatthis kind of centrality can be useful for defining well connectedclusters in the network, and, based on that, for understanding spreadingon the same network.

Our strategy is to choose eigenvector centrality [11] as a usefulmeasure of well-connectedness. Eigenvector centrality (EVC) has thedesirable property that—since it depends on the properties of theneighbourhood of a node, and not just of the node itself—it is rather‘smooth’ over the graph (or network; we use these termsinterchangeably). This is in contrast to the related quantity degreecentrality, which simply counts the links leaving a node and so iscompletely local.

Let us elaborate on this difference. We start with degree centrality. Itmeasures the ‘importance’ or connectedness of a node simply by countingthe node's neighbours. Hence the degree centrality of node i is its nodedegree k_(i). Clearly this quantity is completely local: a given nodemay have a very high degree centrality, and yet all of its neighboursmay have a very low degree centrality—there is no correlation betweenthis quantity from one node to its neighbours. Eigenvector centrality isseemingly (at least, in words) only a slight modification. To find anode's EVC, one (again) counts the node's neighbours. but weighting thecount by the centrality (EVC) of the neighbours. That is: it's not justhow many people you know, but who you know that matters. Mathematicallywe express this by

$\begin{matrix}{e_{i} = {({const}) \times {\sum\limits_{j = {{nn}{(i)}}}{e_{j}.}}}} & (1)\end{matrix}$

Here e_(i) is the EVC of node i, and j=nn(i) means only sum over thenearest neighbours of i. This definition is clearly circular—mycentrality depends on that of my neighbours, but theirs depends also onmine. However Equation (1) is readily solved to find the EVC, as long asone includes the constant (const) in the weighted sum. Furthermore,assuming only that the graph is connected and the links are symmetric,we know that the EVC values will all be positive (although they can be‘practically zero’ for very peripheral nodes).

Thus we see that the EVC depends not only on how many neighbours a nodehas, but also on longer-ranged questions such as how many neighbours anode's neighbours have, etc. In fact, in principle, the EVC of a nodedepends on the whole graph. More relevant for our purposes, however aretwo things: (i) the EVC clearly does measure well-connectedness in somekind of non-local fashion, and (ii) because of (i), the EVC values ofnodes on any given path through the network cannot vary randomly andarbitrarily. That is, Eq. (1) forces the EVC of any node to bepositively coupled to the EVC of that node's neighbours. We like torephrase this as follows: the EVC is ‘smooth’ as one moves over thegraph. (More mathematical arguments for this ‘smoothness’ are given in[1]).

The smoothness of the EVC allows one to think in terms of the‘topography’ of the graph. That is, if a node has high EVC, itsneighbourhood (from smoothness) will also have a somewhat high EVC-sothat one can imagine EVC as a smoothly varying ‘height’, with mountains,valleys, mountaintops, etc. We caution the reader that all standardnotions of topography assume that the rippling ‘surface’ which thetopography describes is continuous (and typically two-dimensional, suchas the Earth's surface). A graph, on the other hand, is not continuous;nor does it (in general) have a clean correspondence with discreteversions of a d-dimensional space for any d. Hence one must usetopographic ideas with care. Nevertheless we will appeal oftentopographic ideas as aids to the intuition. Our definitions will beinspired by this intuition, but still mathematically precise, andappropriate to the realities of a discrete network.

First we define a ‘mountaintop’. This is a point that is higher than allits neighbouring points-a definition which can be applied unchanged tothe case of a discrete network. That is, if a node's EVC is higher thanthat of any of its neighbours (so that it is a local maximum of theEVC), we call that node a Centre. Next, we know that there must be amountain for each mountaintop. We will call these mountains regions; andthey are important entities in our analysis. That is, each node which isnot a Centre must either belong to some Centre's mountain (region), orlie on a ‘border’ between regions. In fact, our preferred definition ofregion membership has essentially no nodes on borders between regions.Thus our definition of regions promises to give us just what we wanted:a way to break up the network into well connected clusters (theregions).

Here is our preferred definition for region membership: all those nodesfor which a steepest-ascent path terminates at the same local maximum ofthe EVC belong to the same region. That is, a given node can find whichregion it belongs to by finding its highest neighbour, and asking thathighest neighbour to find its highest neighbour, and so on, until thesteepest-ascent path terminates at a local maximum of the EVC (i.e., ata Centre). All nodes on that path belong to the region of that Centre.Also, every node will belong to only one Centre, barring the unlikelyevent that a node has two or more highest neighbours having exactly thesame EVC, but belonging to differing regions.

Finally we discuss the idea of ‘valleys’ between regions. Roughlyspeaking, a valley is defined topographically by belonging to neithermountainside that it runs between. Hence, with our definition of regionmembership, essentially no nodes lie in the valleys. Nevertheless it isuseful to think about the ‘space’ between mountains—it is after all this‘space’ that connects the regions, and thus plays an important role inspreading. This ‘valley space’ is however typically composed only ofinter-region links. We call these inter-region links bridging links.(And any node which lies precisely on the border may be termed abridging node.)

FIG. 1 offers a pictorial view of these ideas. We show a simple graphwith 16 nodes. We draw topographic contours of equal height (EVC). Thetwo Centres, and the mountains (regions) associated with each, areclearly visible in the figure. The figure suggests strongly that the tworegions, as defined by our analysis, are better connected internallythan they are to one another. Furthermore, from the figure, it isintuitively plausible that spreading (e.g., of a virus) will occur morereadily within a region than between regions. Hence, FIG. 1 expressespictorially the two aims we seek to achieve by using EVC: (i) find thewell connected clusters, and (ii) understand spreading.

2 Topography and Epidemic Spreading

In order to understand spreading from a network perspective, we wouldlike somehow to evaluate the nodes in a network in terms of their“spreading power”. That is, we know that some nodes play an importantrole in spreading, while others play a less important role. One needonly imagine the extreme case of a star: the centre of the star isabsolutely crucial for spreading of infection over the star; while theleaf nodes are entirely unimportant, having only the one aspect (commonto every node in any network) that they can be infected.

Clearly, the case of the star topology has an obvious answer to thequestion of which nodes have an important role in spreading (have highspreading power). The question is then, how can one generate equallymeaningful answers for general and complex topologies, for which theanswer is not at all obvious? In this section we will propose anddevelop a qualitative answer to this question.

Our basic assumption (A) is simple, and may be expressed in a singlesentence:

Eigenvector Centrality (EVC) is a Good Measure of Spreading Power. (A)

We have tested this idea, via both simulations and theory [2]. Now wewill give qualitative arguments which support assumption (A); we willthen go on to explore the implications of this assumption. We will seethat we can develop a fairly detailed picture of how epidemic spreadingoccurs over a network, based on (A) and our structural analysis—inshort, based on the ideas embodied in FIG. 1.

First we recall that, because a node's EVC depends on that of itsneighbours, the EVC values over a network may be thought of as ‘smoothlyvarying’ over the network. That is, a node with very high EVC cannot besurrounded by nodes with very low EVC. Of course, it is true that EVCtends to be positively correlated with a simpler measure of centrality,namely the node degree. In fact, one might say that the principaldifference between the two measures is that EVC is constrained by itsdefinition to be smooth, while node degree centrality is not [12]. Thisdifference can however be nontrivial. For instance, a node with highdegree, surrounded by many leaf nodes, and linked only tenuously to thebulk of a large and well-connected network, will have a low EVC, inspite of its high degree. The point is that EVC is sensitive toproperties of neighbourhoods, while node degree is not.

Thus, in short, there are no isolated nodes with high EVC. That is, anode with high EVC is embedded in a neighbourhood with high EVC. (Therecan however be relatively isolated nodes with low EVC, as this situationis self-consistent. Low-EVC nodes can be isolated in the sense of havingvery few neighbours; but it is still the case that their neighbours willnot have very much higher EVC.) Now if we take our basic assumption (A)to be true, then there are no isolated nodes with high spreading power.Instead, there are neighbourhoods with high spreading power.

We then suppose that an infection has reached a node with modestspreading power. Suppose further that this node is not a local maximumof EVC; instead, it will have a neighbour or neighbours of even higherspreading power. The same comment applies to these neighbours, until onereaches the local maximum of EVC/spreading power.

Now, given that there are neighbourhoods, we can discuss spreading interms of neighbourhoods rather than in terms of single nodes. It followsfrom the meaning of spreading power that a neighbourhood characterizedby high spreading power will have more rapid spreading than onecharacterized by low spreading power. Furthermore, we note that thesedifferent types of neighbourhoods (high and low) are smoothly joined byareas of intermediate spreading power (and speed).

It follows from all this that, if an infection starts in a neighbourhoodof low spreading power, it will tend to spread to a neighbourhood ofhigher spreading power. That is: spreading is faster towardsneighbourhoods of high spreading power, because spreading is faster insuch neighbourhoods. Then, upon reaching the neighbourhood of thenearest local maximum of spreading power, the infection rate will alsoreach a maximum (with respect to time). Finally, as the highneighbourhood saturates, the infection moves back ‘downhill’, spreadingout in all ‘directions’ from the nearly saturated high neighbourhood,and saturating low neighbourhoods.

We note that this discussion fits naturally with our topographic pictureof network topology. Putting the previous paragraph in this language,then, we get the following: infection of a hillside will tend to moveuphill, while the infection rate grows with height. The top of themountain, once reached, is rapidly infected; and the infected top thenefficiently infects all of the remaining adjoining hillsides. Finally,and at a lower rate, the foot of the mountain is saturated.

FIG. 2 expresses these ideas pictorially. The figure shows ourtwo-region example of FIG. 1, but viewed from the ‘side’—as if each nodetruly has a height. The initial infection occurs at the black node inthe left region. It then spreads primarily uphill, with the rate ofspreading increasing with increasing ‘height’ (=EVC, which tells us, byour assumption, the spreading power). The spreading of the infectionreaches a maximum rate when the most central nodes in the region arereached; it then ‘takes off’, and infects the rest of the region.

We see that this qualitative picture addresses nicely the various stagesof the classic S curve of innovation diffusion [13]. The early, flatpart of the S is the early infection of a low area; during this period,the infection moves uphill, but slowly. The S curve begins to take offas the infection reaches the higher part of the mountain. Then there isa period of rapid growth while the top of the mountain is saturated,along with the neighbouring hillsides. Finally, the infection rate slowsdown again, as the remaining uninfected low-lying areas become infected.

We again summarize these ideas with a figure. FIG. 3 shows a typical Scurve for infection, in the case (as we study in this paper) thatimmunity is not possible. Above this S curve, we plot the expectedcentrality of the newly infected nodes over time. According to ourarguments above, relatively few nodes are infected before the mostcentral node is reached—even as the centrality of the infection front issteadily rising. The takeoff of the infection then roughly coincideswith the infection of the most central neighbourhood. Hence, the part ofFIG. 3 to the left of the dashed line corresponds to the left half ofFIG. 2; similarly, the right-hand parts of the two figures correspond.

One might object that this picture is too simple, in the followingsense. Our picture gives an S curve for a single mountain. Yet we knowthat a network is often composed of several regions (mountains). Thequestion is then, why should such multi-region networks exhibit a singleS curve?

Our answer here is that such networks need not necessarily exhibit asingle S curve. That is, our arguments predict that each region—definedaround a local maximum of the EVC—will have a single S curve.Then—assuming that each node belongs to a single region, as occurs withour preferred rule for region membership—the cumulativeinfection/infection curve for the whole network is simply the sum of theinfection curves for each region. These latter single-region curves willbe S curves. Thus, depending on the relative timing of these varioussingle-region curves, the network as a whole may, or may not, exhibit asingle S curve. For example, if the initial infection is from aperipheral node which is close to only one region, then that region maytake off well before neighbouring regions. On the other hand, if theinitial infection is in a valley which adjoins several mountains, thenthey may all exhibit takeoff roughly simultaneously—with the resultbeing a sum of roughly synchronized S curves, hence a single S curve.

Let us now summarize and enumerate the predictions we take from thisqualitative picture.

-   -   a. Each region has an S curve.    -   b. The number of takeoff/plateau occurrences in the cumulative        curve for the whole network may be more than one; but it will        not be more than the number of regions in the network.    -   c. For each region—assuming (which will be typical) that the        initial infection is not a very central node-growth will at        first be slow.    -   d. For each region (same assumption) initial growth will be        towards higher EVC.    -   e. For each region, when the infection reaches the neighborhood        of high centrality, growth “takes off”.    -   f. An observable consequence of (e) is then that, for each        region, the most central node will be infected at, or after, the        takeoff—but not before.    -   g. For each region, the final stage of growth (saturation) will        be characterized by low centrality.

3 Mathematical Theory

In [2] we have developed a mathematical theory for the qualitative ideasexpressed here. We have focused on two aspects there, which we willsimply summarize here.

Definition of Spreading Power

The first problem is to try to quantify and make precise our assumption(A). Since (A) relates two quantities-spreading power and EVC—and thelatter is precisely defined, the task is then to define the former, andthen to seek a relation between the two.

Such a relation is intuitively reasonable. A node which is connected tomany well-connected nodes should have higher spreading power, and higherEVC, than a node which is connected to equally many, but poorlyconnected, nodes. We have offered a precise definition of spreadingpower in [2]. Our reasoning has two steps: first we define an ‘infectioncoefficient’ C(i,j) between any pair of nodes i and j. This is simply aweighted sum of all non-self-retracing paths between i and j, with lowerweight given to longer paths. Thus many short paths between two nodesgives them a high infection coefficient. Our definition is symmetric, sothat C(i,j)=C(j,i).

Next we define the spreading power of node i to be simply the sum overall other nodes j of its infection coefficient C(i,j) with respect to j.As long as the graph is connected, every node will have a nonzero C(i,j)with every other, thus contributing to the sum. Hence each node has thesame number of terms in the sum; but the nodes with many large infectioncoefficients will of course get a higher spreading power.

We then show in [2] that one can make a strong connection between thisdefinition of spreading power and the EVC, if one can ignore therestriction to non-self-retracing paths in the definition. We restrictthe sum to non-self-retracing paths because self-retracing paths do notcontribute to infection in the SI case. This restriction makes theobtaining of analytical results harder.

Mathematical Theory of SI Spreading

We have given in [2] exact equations for the propagation of aninfection, for arbitrary starting node, in the SI case. These equationsare stochastic—expressed in terms of probabilities—due to theprobabilistic model for spreading over links. They are not generallysolvable, even in the deterministic case when p=1. The problem in thelatter case is again the need to exclude non-self-retracing paths.However we have performed an expansion in powers of p for the timeevolution of the infection probability vector. This expansion shows thatthe dominant terms are those obtained by naively applying the adjacencymatrix (i.e., ignoring self-retracing paths because they are longer,hence higher order in p). The connection to EVC is then made: naivelyapplying the adjacency matrix gives weights (infection probabilities)which approach a distribution proportional to the EVC. Hence we get someconfirmation for our claim that, in the initial stages of an infection,the front moves towards higher EVC.

4 Design and Improvement of Networks

In this section we go beyond the problem of analysis, and address theproblem of design of networks [14]. Our ideas have some clearimplications for design—both towards the aim of preventing the spreadingof harmful information (such as viruses) and towards the aim of helpingspreading—in each case, by modifying the topology of a given network.

Measures to Improve Spreading.

We frame our ideas in terms of our topographic picture. Now we supposethat we wish to design, or modify the design of, a network, so as toimprove its efficiency with respect to spreading. It is reasonable,based on our picture, to assume that a single region is the optimaltopology for efficient spreading. Hence we include, in the presentinvention, four ideas which are expected to improve information flow ina network, by modifying a given (multi-region) network topology to makeit more like a single region:

-   -   1. One can add more bridge links between the regions. Links        between nodes with high EVC in each region are expected to be        most effective. See FIG. 4.    -   2. As an extreme case of 1, one can connect the Centres of the        regions. See FIG. 5.    -   3. One can connect a subset of nodes from different regions by a        relaying star node. See FIG. 10 B.    -   4. One can connect all, or some, of the centre nodes by a        relaying star node. See FIG. 11 B.

Idea 2 is a “greedy” version of idea 1. In fact, the greediest versionof idea 2 is to connect all Centres to all, thus forming a completesub-graph among the Centres. A complete subgraph among 5 Centres isshown in FIG. 11A. An alternative to this design is to insert a new starnode (shown in white in FIG. 11B), which is connected to all the centrenodes by just one link to each. In some situations, where physicallylaying down new links is costly in terms of scarce resources, and addingnew nodes is feasible, this star design can be more attractive than thecomplete sub-graph option. When n Centres are to be connected, the stardesign adds just n new links and one new node, whereas the completesubgraph will add n(n−1)/2 new links. Combinations of these twoapproaches are also possible; one subset of the centres can be connectedas a complete subgraph, whereas a star node can connect another subset.

However, such greedy approaches may in practice be difficult orimpossible. There remain then the general ideas 1 and 3 of building morebridges between the regions. Here we see however no reason for nottaking the greediest practical version of this idea. That is: build thebridges between nodes of high centrality on both sides-preferably, ashigh as possible. Our analysis strongly suggests that this is the beststrategy for modifying topology so as to help spreading. Choosingsubsets of nodes of high EVC score in each region, and then combiningthese subsets, can also be done, as shown in FIG. 10A. Again, in caseswhere one wants to minimize the number of new links added to thenetwork, adding a new star node in between the two subsets of nodes canbe a viable solution. This is shown in FIG. 10B. Connecting all thenodes in one subset with all the nodes in the other subset will requirek*g new links, where k and g are the number of nodes in each subset. Inthe star-node approach the number of added links can be considerablyless: only k+g. Hence, the advantage of the relaying star-node approachshould be clear.

We note that the greediest strategy is almost guaranteed to give asingle-region topology (and therefore efficient information spreading)as a result. Our reasoning is simple. First, the existing Centres cannotall be Centres after they are all connected one to another-because twoadjacent nodes cannot both be local maxima of the EVC (or of anythingelse). Therefore, either new Centres turn up among the remaining nodesas a result of the topology modification, or only one Centre survivesthe modification. In the latter case we have one region. The formercase, we argue, is unlikely: we note that the EVC of the existingCentres is (plausibly) strengthened (raised) by the modification morethan the EVC of other nodes. That is, we believe that connectingexisting centres in a complete sub-graph will ‘lift them up’ withrespect to the other nodes, as well as bringing them closer together. Ifthis ‘lifting’ idea is correct, then we end up with a single Centre anda single region.

Measures to Prevent Spreading

Now we address the problem of designing, or redesigning, a networktopology so as to hinder spreading. Here the problem is more complicatedthan in the helping case. The reason for this is that we build networksin order to support and facilitate communication. Hence we cannot simplyseek the extreme, ‘perfect’ solution-because the ideal solution forhindering spreading is one region per node, i.e., disconnect all nodesfrom all others! Instead we must consider incremental changes to a givennetwork. We consider two types of ‘inoculation’ strategies: inoculatingnodes (which is equivalent to removing them, as far as spreading isconcerned), or inoculating links (which is also equivalent to removingthem). Again we include in the present invention a list of ideas, nowuseful for hindering spreading:

-   -   1. One can inoculate the Centres (see FIG. 6)—along with,        perhaps, a small neighbourhood around them.    -   2 One can instead find a ring of nodes surrounding each Centre        (at a radius of perhaps two or three hops) and inoculate the        ring. In FIG. 7, a ring of nodes at one hop from each Centre is        inoculated.    -   3. One can inoculate bridge links. See FIG. 9.    -   4. One can inoculate nodes at the ends of bridge links. See FIG.        8.

We note that ideas 1 and 2 are applicable even in the case that only asingle region is present. Ideas 3 and 4 may be used when multipleregions are found. Note that inoculating a bridge link (idea 3) is notthe same as inoculating the two nodes which the link joins (idea 4):inoculating a node effectively removes that node and all links connectedto it, while inoculating a link removes only that link. In FIG. 8, thetwo nodes at the ends of the bridge link are inoculated, while in FIG.9, only the bridge link itself is inoculated.

Also, with link inoculation, one has the same considerations as withlink addition-namely, the height of the link matters. We define the“link EVC” to be the arithmetic mean of the EVC values of the nodes onthe ends of the link. Ideas 3 and 4 are then almost certainly mosteffective if the bridging links chosen for inoculation have a relativelyhigh link EVC.

Inoculating a link means removing the link. And “removing” meansblocking any and all communication over the link. Now, given thisdefinition, we can say that inoculating a node means inoculating ALLlinks connected to that node. In this way, no communication to or fromthe inoculated node is possible. This is equivalent to “removing thenode from the graph”. For our purposes, it is not necessary to shut downa node in order to inoculate it. One must simply close off allcommunication to & from the node.

Another definition of inoculation is possible. If it is possible todetect and block the unwanted information, and thereby to filter thecommunication over links in some way, then we need not close off ALLcommunication on a link in order to inoculate the link. That is, if wecan detect the unwanted, harmful communication (e.g., a virus), then itis sufficient to block only THAT form for communication, and allow othercommunications through. Inoculation of a link may then be defined as:Blocking any transmission of “unwanted” information over the link. Theninoculating a node can be defined as inoculating all links connected tothe node (as before).

REFERENCES

-   [1] Geoffrey Canright and Kenth Engø-Monsen, “Roles in Networks”.    Science of Computer Programming, 53 (2004) 195-214.-   [2] Geoffrey Canright and Kenth Engo-Monsen, “Spreading on networks:    a topographic view”, submitted to European Conference on Complex    Systems (ECCS05).-   [3] Geoffrey Canright, Kenth Engo-Monsen, Asmund Weltzien, and    Fahimeh Pourbayat, “Diffusion in social networks and disruptive    innovations”. IADIS e-Commerce 2004 proceedings. Lisbon 2004.-   [4] M. E. J. Newman, “The structure and function of complex    networks”. SIAM Review, 45 (2003), 167-256.-   [5] Romualdo Pastor-Satorras and Alessandro Vespignani, “Epidemic    Spreading in Scale-Free Networks”. Phys. Rev. Lett 86 (2001),    3200-3203.-   [6] Romualdo Pastor-Satorras and Alessandro Vespignani, “Epidemic    dynamics and endemic states in complex networks”. Phys. Rev. E 63,    066117 (2001).-   [7] M. E. J. Newman, “Spread of epidemic disease on networks”. Phys.    Rev. E 66, 016128 (2002).-   [8] Fred Brauer, “A model for an SI disease in an age-structured    population”. Discrete and Continuous Dynamical Systems B2 (2002),    257-264.-   [9] Yang Wang, Deepayan Chakrabarti, Chenxi Wang, and Christos    Faloutsos, “Epidemic spreading in real networks: an eigenvalue    viewpoint”. Proceedings, 22^(nd) Symposium on Reliable Distributed    Systems (SRDS 2003), 25-34.-   [10] A good introduction to many of these definitions may be found    in: http://www.analytictech.com/networks/centrali.htm-   [11] P. Bonacich, “Factoring and weighting approaches to status    scores and clique identification”. Journal of Mathematical    Sociology, 2 (1972), 113-120.-   [12] The star illustrates this difference to some extent. Suppose    the graph is a star with n ‘leaves’—that is, a graph with one node    in the center, linked to each of n other nodes, each of which have    no neighbour other than the center node. The degree centrality of    the center is of course n, and that of the leaves is 1. The EVC of    the center is however only √{square root over (n)} larger than the    EVC of the leaves. Hence using EVC-which makes the centrality of the    center dependent on that of its neighbors-gives a reduction (by a    factor 1/√{square root over (n)}) in the (potentially large)    difference in degree centrality between leaves and center.-   [13] E. M. Rogers, Diffusion of Innovations, 3^(rd) ed. Free Press,    New York (1983).-   [14] For a discussion of closely related ideas, see: M. Burgess, G.    Canright, and K. Engø. “A graph theoretical model of computer    security: from file access to social engineering”. International    Journal of Information Security, Volume 3, Number 2, November 2004,    pages 70-85.

1-12. (canceled)
 13. A method of spreading information in a network,said network including a plurality of network nodes interconnected bylinks, said method including steps of: (a) mapping a topology of saidplurality of nodes of said network; (b) computing one or more values oflink strength between said nodes; (c) computing Eigenvector CentralityIndexes for said plurality of nodes, said Indexes being computed fromsaid one or more values of link strength; (d) identifying nodes whichare local maxima of said Indexes, said identified nodes being denoted tobe centre nodes; (e) identifying regions associated with said centrenodes, and associating said centre nodes with corresponding networknodes in their corresponding regions, characterized in that said methodincludes further steps of: (f) identifying for each region correspondingnodes into first and second groups of nodes, said first group of nodesincluding said centre node of said region, and said second group ofnodes having Eigenvector Centrality Indexes less than nodes included insaid first group; and (g) connecting at least one network node of afirst group of a first region to at least one network node of a firstgroup of a second region.
 14. A method as claimed in claim 13,characterized in that said method includes a step of connecting thecentre node of said first region with all or a subset of the centrenodes of said multiple regions into a complete graph.
 15. A method asclaimed in claim 13, characterized in that said method includes a stepof connecting all or a subset of EVC nodes from said first group of saidfirst region, with all or a subset of EVC nodes of the first group ofsaid second region.
 16. A method of spreading information in a network,said network including a plurality of network nodes interconnected bylinks, said method including steps of: (a) mapping a topology of saidplurality of nodes of said network; (b) computing one or more values oflink strength between said nodes; (c) computing Eigenvector CentralityIndexes for said plurality of nodes, said Indexes being computed fromsaid one or more values of link strength; (d) identifying nodes whichare local maxima of said Indexes, said identified nodes being denoted tobe centre nodes; (e) identifying regions associated with said centrenodes, and associating said centre nodes with corresponding networknodes in their corresponding regions, characterized in that said methodincludes further steps of: (f) identifying for each region correspondingnodes into first and second groups of nodes, said first group of nodesincluding said centre node of said region, and said second group ofnodes having Eigenvector Centrality Indexes less than nodes included insaid first group; and (g) adding at least one new node, and connectingat least one existing node from the first group in first and secondregions with said at least one new node.
 17. A method as claimed inclaim 16, characterized in said method includes a step of connectingwith a direct link all or a subset of the centre nodes of said multipleregions to said new node, thus forming a star graph.
 18. A method asclaimed in claim 16, characterized in said method includes a step ofconnecting all or a subset of EVC nodes of said first group of saidfirst and second regions to said new node, thus forming a star graph.19. A method of preventing spreading of information in a network, saidnetwork including a plurality of network nodes interconnected by links,said method including steps of: (a) mapping a topology of said pluralityof nodes of said network; (b) computing one or more values of linkstrength between said nodes; (c) computing Eigenvector CentralityIndexes for said plurality of nodes, said Indexes being computed fromsaid one or more values of link strength; (d) identifying nodes whichare local maxima of said Indexes, said identified nodes being denoted tobe centre nodes; (e) identifying regions associated with said centrenodes, and associating said centre nodes with corresponding networknodes in their corresponding regions, characterized in that said methodincludes further steps of: (f) identifying for each region correspondingnodes into first and second groups of nodes, said first group of nodesincluding said centre node of said region, and said second group ofnodes having Eigenvector Centrality Indexes less than nodes included insaid first group; and (g) inoculating at least one node of EigenvectorCentrality Index from said first group by blocking any transmission ofunwanted information on all links in/out of said node.
 20. A method asclaimed in claim 19, characterized in that said method includes a stepof inoculating all or a subset of the centre nodes.
 21. A method asclaimed in claim 19, characterized in that said method includes a stepof inoculating a ring of nodes at a distance of hops of at least one hopaway from the said centre node.
 22. A method as claimed in claim 19,characterized in that said method includes a step of inoculating all ora subset of bridge nodes of said first groups of said regions.
 23. Amethod for preventing spreading of information in a network, saidnetwork including a plurality of network nodes interconnected by links,said method including steps of: (a) mapping a topology of said pluralityof nodes of said network; (b) computing one or more values of linkstrength between said nodes; (c) computing Eigenvector CentralityIndexes for said plurality of nodes, said Indexes being computed fromsaid one or more values of link strength; (d) identifying nodes whichare local maxima of said Indexes, said identified nodes being denoted tobe centre nodes; (e) identifying regions associated with said centrenodes, and associating said centre nodes with corresponding networknodes in their corresponding regions, characterized in that said methodincludes further steps of: (f) identifying for each region correspondingnodes into first and second groups of nodes, said first group of nodesincluding said centre node of said region, and said second group ofnodes having Eigenvector Centrality Indexes less than nodes included insaid first group; (g) identifying links which connect two regions asbridging links; (h) identifying bridging links with EigenvectorCentrality nodes from first groups at each end of said links as highEigenvector Centrality links; and (i) inoculating at least one highEigenvector Centrality link by blocking any transmission of unwantedinformation on said Eigenvector Centrality link.
 24. A method as claimedin claim 23, characterized in that said method includes a step ofinoculating all or a subset of bridge links of high link centrality.